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Tridiagonal matrix algorithm - TDMA (Thomas algorithm)

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Introduction

The Thomas Algorithm is a special form of Gaussian elimination that can be used to solve tridiagonal systems of equations. When the matrix is tridiagonal, the solution can be obtained in O(n) operations, instead of O(n^3). Example of such matrices are matrices arising from discretization of 1D problems.

We can write the tri-diagonal system in the form:

a_i x_{i - 1} + b_i x_i + c_i x_{i + 1} = d_i

Where a_1 = 0 and c_n = 0

Algorithm

for k:= 2 step until n do
m = {{a_k } \over {b_{k - 1} }}
b_k^' = b_k - mc_{k - 1}
d_k^' = d_k - md_{k - 1}
end loop (k)
then
\phi_n = {{d_n^' } \over {b_n }}
for k := n-1 stepdown until 1 do
\phi_k = {{d_k^' - c_k \phi_{k + 1} } \over {b_k }}
end loop (k)

References

  1. Conte, S.D., and deBoor, C. (1972), Elementary Numerical Analysis, McGraw-Hill, New York..


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